A concave mirror is a converging mirror and can form real images of real objects when the object is beyond the focal point. Therefore, both Assertion (A) and Reason (R) are false.

For a concave mirror, \(R = 20 \text{cm}\), so \(f = R/2 = 10 \text{cm}\). An object placed at the focal point of a converging mirror forms its image at infinity. Both A and R are true, and R correctly explains A.

A large diameter objective lens in telescopes increases the light-gathering power and improves resolving power, allowing observation of fainter and finer details. Hence, Assertion (A) is true. However, spherical aberration generally *increases* with larger apertures, as rays further from the principal axis are not focused at the same point. Thus, Reason (R) is false.

For a concave mirror, if an object moves from infinity towards the center of curvature (C), its real image moves from the focus (F) to C, thus moving *towards* the mirror. So, Assertion (A) is false. Also, the object and image do not *always* travel in the same direction along the principal axis; their relative motions depend on the object's position relative to F and C. Hence, Reason (R) is also false. Both A and R are false.

Convex mirrors are used as rear-view mirrors because they provide a wider field of view and always form virtual, erect, and diminished images of real objects. This property is crucial for drivers to see a larger area behind the vehicle and perceive objects as being further away. Both A and R are true, and R correctly explains the suitability of convex mirrors for this application.

Covering half a mirror reduces the intensity of the image but a complete image is still formed. The focal length of a mirror depends only on its radius of curvature, not on the aperture or how much of its surface is exposed.

Both Assertion (A) and Reason (R) are false. A concave mirror can form a virtual image of a virtual object if the object is between its pole and focus. A convex mirror can form a real image of a virtual object, for example, if the virtual object is placed between its pole and focus.

For a concave mirror, the image acceleration is not uniform, as magnification \(m = -v/u\) changes non-linearly with object position (u).
Thus, (A) is false.
A concave mirror is a converging mirror. It acts as a diverging mirror only when the object is placed between the pole and focus, forming a virtual, erect, and magnified image. However, in general context, it's not a diverging mirror. Thus, (R) is also false.
Therefore, both (A) and (R) are false.

The focal length of a spherical mirror ((f = R/2)) depends only on its geometric curvature, not on the wavelength of light or the medium's refractive index. Thus, (A) is true.
When light reflects from the surface of an optically denser medium (e.g., from air to glass), there is a phase change of \(\pi\) (or \(180^\circ)\). Thus, (R) is true.
However, the phase change on reflection does not explain why the focal length of a mirror is independent of wavelength. This independence is due to the nature of reflection itself, where the angle of incidence equals the angle of reflection regardless of wavelength.

If an object moves along the principal axis, its image also moves along it, with velocities always anti-parallel (angle \(180^\circ\)), so (A) is true. Perpendicular components of velocity are related by \(v_{iy} = m v_{oy}\), where magnification \(m\) is generally not 1, so (R) is true. (R) does not explain (A).